Acceleration (differential geometry)

In mathematics and physics, acceleration is the rate of change of velocity of a curve with respect to a given linear connection. This operation provides us with a measure of the rate and direction of the "bend".

Formal definition

Consider a differentiable manifold $M$ with a given connection $\mathrm{\Gamma }$. Let $\gamma :\mathbb{ℝ}\to M$ be a curve in $M$ with tangent vector, i.e. velocity, $\dot{\gamma }(\tau )$, with parameter $\tau$.

The acceleration vector of $\gamma$ is defined by ${\mathrm{\nabla }}_{\dot{\gamma }}\dot{\gamma }$, where $\mathrm{\nabla }$ denotes the covariant derivative associated to $\mathrm{\Gamma }$.

It is a covariant derivative along $\gamma$, and it is often denoted by

${\mathrm{\nabla }}_{\dot{\gamma }}\dot{\gamma }=\frac{\mathrm{\nabla }\dot{\gamma }}{d\tau }.$

With respect to an arbitrary coordinate system $(x^{\mu })$, and with $({\mathrm{\Gamma }}^{\lambda }{}_{\mu \nu })$ being the components of the connection (i.e., covariant derivative ${\mathrm{\nabla }}_{\mu }:={\mathrm{\nabla }}_{\partial /\partial x^{\mu }}$) relative to this coordinate system, defined by

${\mathrm{\nabla }}_{\partial /\partial x^{\mu }}\frac{\partial }{\partial x^{\nu }}={\mathrm{\Gamma }}^{\lambda }{}_{\mu \nu }\frac{\partial }{\partial x^{\lambda }},$

for the acceleration vector field $a^{\mu }:=({\mathrm{\nabla }}_{\dot{\gamma }}\dot{\gamma }{)}^{\mu }$ one gets:

$a^{\mu }=v^{\rho }{\mathrm{\nabla }}_{\rho }{v}^{\mu }=\frac{d{v}^{\mu }}{d\tau }+{\mathrm{\Gamma }}^{\mu }{}_{\nu \lambda }{v}^{\nu }{v}^{\lambda }=\frac{d^2{x}^{\mu }}{d{\tau }^2}+{\mathrm{\Gamma }}^{\mu }{}_{\nu \lambda }\frac{d{x}^{\nu }}{d\tau }\frac{d{x}^{\lambda }}{d\tau },$

where $x^{\mu }(\tau ):={\gamma }^{\mu }(\tau )$ is the local expression for the path $\gamma$, and $v^{\rho }:=(\dot{\gamma }{)}^{\rho }$.

The concept of acceleration is a covariant derivative concept. In other words, in order to define acceleration an additional structure on $M$ must be given.

Using abstract index notation, the acceleration of a given curve with unit tangent vector ${\xi }^a$ is given by ${\xi }^b{\mathrm{\nabla }}_b{\xi }^a$.

See also

• Acceleration
• Covariant derivative

References

• Friedman, M. (1983). Foundations of Space-Time Theories. Princeton: Princeton University Press. ISBN 0-691-07239-6. 
• Dillen, F. J. E.; Verstraelen, L.C.A. (2000). Handbook of Differential Geometry. 1. Amsterdam: North-Holland. ISBN 0-444-82240-2. 
• Pfister, Herbert; King, Markus (2015). Inertia and Gravitation. The Fundamental Nature and Structure of Space-Time. The Lecture Notes in Physics. Volume 897. Heidelberg: Springer. doi:10.1007/978-3-319-15036-9. ISBN 978-3-319-15035-2.